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GRAPH TOPOLOGY FOR FUNCTION SPACES(')
BY
SOMASHEKHAR AMR1TH NAIMPALLY
1. Introduction. Most of the work in function space topologies concerns
continuous functions. In this connection see a remark by Kelley [3, p. 217]. As
soon as we begin to consider function spaces of noncontinuous functions we
come face to face with some extremely difficult problems. So in order to make
a beginning, it is advisable to consider first a subfamily of noncontinuous functions which, in a certain sense, can be approximated by continuous functions.
One such subfamily consists of almost continuous functions which were introduced
by Stallings [6]. An almost continuous function is one whose graph can be
approximated by graphs of continuous functions (see 2.3). The need to introduce
a suitable topology for the function space of almost continuous functions arose
when the author was investigating the essential fixed points of such functions
in his doctoral thesis [4]. The introduction of a new function space topology,
called "the graph topology", enabled him to tackle almost continuous functions.
Let F denote an arbitrary subfamily of functions on a topological space X
to a topological space Y and let F be given some topology. Most problems concerning F center round the following question, "what conditions on X and Y
are sufficient to ensure that F has a desired property?" In this paper a few problems of the above nature are discussed.
This paper has a nonempty intersection with the author's doctoral thesis
written under the supervision of Professor J. G. Hocking of Michigan State
University. The author is grateful to his former colleague Professor D. E. Sanderson for valuable suggestions and comments. The referee suggested several
improvements, supplied Example 5.1 and the references [1] and [5].
2. Graph topology.
2.1. Let X and Y be topological spaces and let F denote the set of all functions
on X to Y. Let C denote the subset of F consisting of all continuous functions.
2.2. For/ e£, the graph of/, denoted by G(f), is the set
{(x,/(x))|xeA}cAx
F.
Let A x F be assigned the usual product topology.
Presented to the Society, January 27, 1965; received by the editors October 30, 1964
and, in revised form, January 12, 1966.
0) This paper was completed when the author was at Iowa State University, Ames, Iowa.
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S. A. NAIMPALLY
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2.3. A function feF is called almost continuous iff for each open set U in
X x F containing G(f), there exists a geC such that G(g) c U (Stallings [6]).
2.4. Let A he the subset of £ consisting of all almost continuous functions.
Whereas every continuous function is almost continuous, there exist almost
continuous functions which are not continuous. If X = Y = the set of all real
numbers with the usual topology, then the function/ e£ defined by
f(x) —sin = 0
for x / 0
for x = 0,
is almost continuous but not continuous.
2.5. Corresponding to each open set U in X x Y, let Fv = {/e£| G(/)<=[/}.
The topology induced on £ by a basis consisting of sets of the form {Fv} for
each open set U in X x Y is called the graph topology F for £. Let Au = FVC\A
and Cv = FunC.
Even a casual glance at Chapter 7 on Function Spaces in Kelley [3] shows
that the properties of £ under the usual function space topologies, lean rather
heavily on the topology of Y and that the topology of X does not enter into
the definitions or the theorems. In contrast to this it will be seen that, in general,
the properties of £ under F depend on the topologies of both X and Y, and so
in a certain sense T is a "natural" topology for £.
When F is a uniform space the uniform convergence topology for £ is such
that the family C of functions continuous relative to a topology for X is closed
in £. Compare this result with the following theorem, the proof of which is
obvious.
2.6. Theorem. The family of almost continuous functions on X to Y is
closed in (£, T); in fact, it is the closure of the set C of continuous functions.
3. Separation properties.
3.1. Example.
Let X consist of two points a, b with the topology consisting
of 0, {a}, {a, b}. Let Y be the discrete topological space formed by two points
p, q . Let /, geF such that/(a)
= p, f(b) = g(a) = g(b) = q . Any open set
in X x Y containing G(f) also contains G(g) and so (£, T) is not even Tt although y is a metrizable space. This is in contrast to the usual function space
topologies (Kelley [3, pp. 217-237]), whose separation properties Tx, T2 are
inherited from the corresponding ones for Y irrespective of the topology
for X. Incidentally / is almost continuous, but is not even a connected
function.
3.2. Theorem. If Y contains at least two points, then the following
equivalent: (i) X and Y are Tx-spaces> (ii)(£, T) is Tlt (iii)(i4, T) is Tx\
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are
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GRAPH TOPOLOGYFOR FUNCTION SPACES
269
Proof. First let A and Y be ^-spaces and let/, geF and/#
g. Then there
exists an a e X such that f(a) # g(a). Since Y is Ty there exists an open set U
in Y such that/(a) e U but g(a) $ U. Also since A is Ty, (X — {a}) x Y is open
in A x Y. Consequently, Ax U U(A - {a}) x Y contains G(f) but not
G(g). So (F,F) is Ty.
Next let (F, F) be Ty. If F is not Ty, there exist distinct points p, qe y such
that every open set which contains p also contains q. Let/, geF be such that
f(x) = p,g(x) = q for all x e A. Then every open set in A x Y which contains G(f) also
containsG(g)andso£isnot
T,, a contradiction. Thus Y is T¡. To show that X is T,,
assume this is false. Then there exist a, be A such that each open set in A"
containing a also contains b. Let p,q be two distinct points of Y. Define/, geF
such that f(b) = a, g(b) = P, f(x) = g(x) = p for all x e A, x # b. Then every
open set in A x Y which contains G(f) also contains G(g) and so (£, T) is not
Ty, a contradiction.
This shows the equivalence of (i) and (ii). Their equivalence with (iii) follows
by noting that /'s and g's constructed are in fact almost continuous.
We can similarly prove the following theorem.
3.3. Theorem.
// Y has at least two points, then (F, F) is T2 if and only
ifX is Ty and Y is T2 .
4. Graph topology and other function space topologies. In this section we
compare the graph topology with some of the other function space topologies
such as the pointwise convergence (p.c.) topology, the compact-open or /c-topology,
the uniform convergence (u.c.) topology. Recall that in Example 3.1 (F, F) is
not Ty whereas £ is discrete in all the other topologies mentioned above.
This together with Theorem 3.2 shows that in order to get meaningful results
it would be desirable to have A a Trspace.
4.1. Theorem. If X is a Ty-space then the p.c. topology is contained
the graph topology.
in
Proof. For a e A and U an open subset of Y, let W(a, U) = {feF \f(a) e U}.
The sets {W(a, U)} for each a e X and U an open subset of Y form a subbasis
for the p.c. topology of £. Since A is Ty, the set F = (A x U) U (X - {a}) x Y
is open in A x Yand clearly Fv = W(a, U). Thus W(a, U) is open in F.
4.2. Theorem.
// A is a T2-space then the graph topology contains the
k-topology. If further X is compact, the graph topology coincides with the
k-topology.
Proof. A subbasis for the /c-topology for £ consists of sets of the form
W(K, U) = {fe F | f(K) c V} for each compact set K in X and each open set
U in Y. Since X is T2 the set V=(XxU)
V(X-K)x
Y is open in A"x Y and
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S. A. NAIMPALLY
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clearly Fv = W(K, U). This proves the first statement and the proof of the
second statement is trivial.
4.3. Example.
We now give an example to show that, in general, the
/c-topology is properly contained in the graph topology. Let X = Y — the set
of all real numbers with the usual topology. Let/e£
be such that/(x) = x for
all x e X. Let U be the open set in X x Y, which is the union of all open discs
with centers at the points of G(f) and radius e > 0. Then G(f) cz U.If
Klt K2, •••, Kn are compact subsets of X and Uu U2, —, U„ are open subsets
of y such that f(K¡) cz U¡, i = 1, 2, •••, n, then f ef],n=1 W(K„ l/(). Let
peX, be such that p£ U?=i K¡. Define geF such that g(x) =f(x) for all
xeX,
x * p and g(p) = p + 2e. Then gef)Ui
W(K„ U¡) but G(g) dz U.
Thus Fv is not open in the /c-topology. The above construction can easily be
altered to make g continuous, showing that the /c-topology # the graph topology,
even on the subspace C.
The first part of the following theorem is obvious and the proof of the second
part is similar to that of Example 4.3.
4.4. Theorem.
The p.c. topology cz the k-topology cz the graph topology
when X is T2. Moreover, ifX is not compact and the topology of Y is not trivial
(indiscrete), the last inclusion is not reversible.
4.5. Theorem. Let X and Y be uniform spaces with uniformities <%and V
respectively. Let F'czF consist of functions which are uniformly continuous
relative to <%and V. Then the u. c. topology for £' is contained in the graph
topology.
Proof. A basis for the u.c. uniformity for £' consists of sets of the form
W(V) = {(/, g) e£' x £'|(/(x),
g(x))e V e"T for all x e X). Consider
W(V) [/] where /e£',
Ver.
Let V2eT be such that F2oF2c F. Since
/g£',
corresponding to F2 there exists a VxeW such that for all p,qeX,
(p, q) e Vt implies (f(p), f(q)) e V2. We may, without any loss of generality,
suppose that Vu V2 are symmetric. Then U = \JxeX {Fi[x] x F2[/(x)]} is
an open set in X x Y containing G(f). Let geF'v = £' r\Fv i.e. G(g) cz U.
For an arbitrary peX, there exists a q e X such that (p, g(p)) e Fj [aj x F2 [f(q)~].
But then f(q) e F2[/(p)] and so g(p) e V2o F2[/(p)] cr F[/(p)].
This
shows that F'v c W(V) [/] i.e. W(V) [/] is open in (£', T).
4.6. Theorem. Let X and Y be uniform spaces with uniformities % and "T
respectively. If X is compact then the u.c. topology is equivalent to the graph
topologyfor C.
Proof. Each/eC is uniformly continuous since X is compact and so in view
of Theorem 4.6, it is sufficient to prove that the graph topology is contained
in the u.c. topology for C. Let/e C and let U be an open set in X x Y containing
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GRAPH TOPOLOGYFOR FUNCTION SPACES
G(f). Since fe C,
there exists a ^st
271
G(f) is homeomorphic to X and so G(f) is compact. Hence
and a V2ef
such that {JxeX {F,[x] x F2[/(x)]} c U
(see Kelley [3, p. 199]). Now if geC and ge W(V2) [/] then g(x)eV2 [/(*)]
for all x e A and so G(g) a U. This shows that C is open in the u.c. topology
forC.
Finally we consider metric spaces. Let A and Y be metric spaces with bounded
metrics d, d' respectively and let A x Y be assigned the product metric D defined by D((xy, yy), (x2, y2)) = d(xy,x2) + d'(yy,y2) where Xy,x2eX
and
ylf y2e Y. Let H be the Hausdorff metric (see Kelley [3, p. 131]) on the hyperspace of all nonempty closed subsets of X x Y. We introduce a metric p in
£ as follows, p(f, g) = H (G(f), G(g)) where/, geF and the bar denotes the
closure operator. Clearly p is a pseudometric for F but we can make £ a metric
space by agreeing that /~ g iff G(f) = G(g) for f, geF and then passing on
to the quotient space with respect to this equivalence relation.
Under the metric p the elements of C still retain their identity since for each
fe C, G(f) is closed in A x Y. In fact, we show below that p is equivalent
to the "supremum metric" py which is usually introduced C where,
Pi (f, g) = sup d'(f(x), g(x)), f, geC.
xeX
4.7. Theorem.
// A is compact then the metrics p and py are equivalent for C.
Proof. Let e>0 and let for feC, U(f, s) = {g eC|p(/, g) < e} and
Uy(f,e) = {geC\Py(f,g)<e}.
Clearly Uy(f, s) a U(f, e). We now show
that there exists a 8 > 0 such that U(f, 8) cz Uy(f, e). Since/e C and A"is compact,
there exists a 8 > 0 (8 < e/3) such that for all x, ye A, d(x, y) <8 implies
d'(f(x), f(y)) < £/3. If geU(f,8)
and aeX
then there exists a be A" such
that d(a, b) + d'(g(a),f(b)) < 8. Thus d'(f(a), g(a))= d'(f(a), f(b)) + d'(f(b),
g(a)) <sß + 8. This shows that U(f, 8) c Ut(f, e).
Similarly we can prove the following theorem.
4.8. Theorem.
Under the hypothesis of Theorem 4.1, the graph topology is
equivalent to the p-metric topology for C.
Analogs of Theorems 4.7 and 4.8 for A would be false even when A, Y are
closed linear intervals, no two of p, px, F need agree on A. This can be shown
by using examples similar to 2.4.
5. Connectedness. An interesting question is "if X and Y are connected
is (A, F) connected?" For noncompact spaces the following counter example
was supplied by the referee.
5.1. Example. De Groot [1] has shown that there exist nondegenerate
connected subsets S, T of the plane such that evety continuous image of either
in the other is a single point. Define a plane set A as follows. Let P„ denote the
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272
S. A. NAIMPALLY
point (n, o), where neN,
the set of all integers. For each neN,
join P2n to P2„+i
by a copy Sn of SOying, except for P2„ and P2n+1, in the slab {(x, y) 12n<x <2n+J),
and P2„_1 to P2„ by a copy T„ of T(lying likewise between these two points).
Let X = (J {S,, U T„ | n e A/}. Let £ = set of all almost continuous maps / of X
in X such that/(P„) = P„ for all neN. It is easy see that £ is closed in (A, F).
Also it is open. For let F„ denote the open neighborhood
{z|zeX,
p(P„, z)<l}
of P„ in X, put U = {(X - N) x X} U {F„ x V„\neN}, an open subset of
XxX,
and If = {g|ge£,
G(g)<=l/}; we show that WnA = E. Clearly
£ c If nX,
WnCczE.
so it suffices (since IF is open and £ is closed and A = C) to prove
Suppose fe W n C ; then/(P„) e F„, for all neN.
Fix n for the
present, and suppose first that n is even, = 2m. If/(P„) is to the left of P„, it
is in Tm. There is a retraction r of X onto Tm (mapping all points to the right
of P„ onto P„, and all points to the left of Pn-y onto P„_i) ; r(f(Sm)) is a continuous image of Sm in Tm, hence a single point. But it contains both r(f(Pn))
and r(f(Pn+ J) = P„ ; so f(P„) = P„. Similar arguments apply if f(Pn) is to the
right of P„, or if n is odd. Thus/(P„) = P„ for all neN, and/e£,
as required.
Clearly 0 # £ 5¿ ^4, so A is not connected.
The question is still open for compact spaces.(2)
Two remarks are in order here. From Theorem 2.6 it follows that if (C, T)
is connected then so is (A, T). It is well known (see Hocking and Young
[2, p. 155], for example) that if Fis a contractible T2 space then C is arcwise
connected. So if X is compact T2 and Y a contractible T2 space then (A, T) is
connected . (A, T) is also connected if X is contractible and Fis arcwise connected,
since every continuous/: X -* Y is homotopic to a constant map.
References
1. J. de Groot, Example of two sets neither of which contains a continuous image of the
other, Indag. Math. 16 (1954),525-526.
2. J. G. Hocking and G. S. Young, Topology, Addison-Wesley, Reading, Mass., 1961.
3. J. L. Kelley, General topology, Van Nostrand, Princeton, N. J., 1955.
4. S. A. Naimpally, Essential fixed points and almost continuous functions, Ph. D. Thesis,
Michigan State University, East Lansing, 1964.
5. V. I. Ponomarev, A new space of closed sets and many-valued functions,
Dokl. Akad.
Nauk SSSR.118(1958),1081-1084.
6. J. R. Stallings, Fixedpoint theorems for connectivitymaps, Fund. Math. 47(1959), 249-263.
University of Alberta,
Edmonton, Canada
(2) Professor D. E. Sanderson has communicated a solution of this problem.
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